Is it possible to generalize it?

  • Thread starter sukyung
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In summary, generalization is a widely accepted scientific method that involves using a sample to draw conclusions and make predictions about a larger population. It differs from extrapolation, which involves making predictions based on data from outside the population. Generalization can be used in all fields of science, but it has limitations such as the need for a representative sample and the possibility of oversimplification. To improve generalization, it is important to ensure a representative sample and use advanced statistical methods, as well as conducting further research.
  • #1
sukyung
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Denote Z_n=(0.1.2...n-1)

Then could I generalize the number of homomorphism H:Z_n -> Z_m as

gcd(n, m)=#(H:Z_n -> Z_m) ?

(Don't consider the case H:Z -> Z)

For example #(H: Z_4 -> Z_2)=2
#(H: Z_12 -> Z_5)= 1 (obviously the trivial one)
 
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  • #2
Yes, and in fact [tex]Hom_\mathbb{Z}(Z_n,Z_m) \cong Z_{(n,m)}[/tex], that is, the homomorphisms from Z_n to Z_m form a cyclic group of order gcd(n,m).
 

1. Is generalization a valid scientific method?

Yes, generalization is a widely accepted scientific method used to draw conclusions and make predictions based on a sample or subset of a larger population.

2. How is generalization different from extrapolation?

Generalization involves taking a sample from a specific population and applying the results to the entire population. Extrapolation, on the other hand, involves making predictions or inferences about a population based on data from outside the population.

3. Can generalization be used in all fields of science?

Yes, generalization can be used in all fields of science as long as there is a representative sample and proper statistical methods are used to draw conclusions.

4. What are the limitations of generalization?

One limitation of generalization is that it relies on the assumption that the sample is representative of the entire population. If the sample is biased or not truly representative, the conclusions drawn may not be accurate. Additionally, generalization can also lead to oversimplification of complex phenomena.

5. How can generalization be improved?

Generalization can be improved by ensuring the sample is truly representative of the population and by using more advanced statistical methods to analyze the data. Additionally, incorporating multiple samples and conducting further research can help to strengthen the validity of generalizations.

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